Constitutive theory for mechanics of amorphous thermoplastic polymers under extreme dynamic loading

A geometrically nonlinear continuum mechanical theory is formulated for deformation and failure behaviors of amorphous polymers. The model seeks to capture material response over a range of loading rates, temperatures, and stress states encompassing shock compression, inelasticity, melting, decomposition, and spallation. Thermoelasticity, viscoelasticity, viscoplasticity, ductile failure with localized shear yielding, and brittle fracture with crazing can all emerge under this ensemble of intense loading conditions. Known prior theories have considered one or more, but not all, such physical mechanisms. The present coherent formulation invokes thermodynamics with internal state variables for dynamic molecular and network configurational changes affecting viscoelasticity and plastic deformation, and it uses order parameters for more abrupt structural changes across state-dependent glass-transition and shock-decomposition thresholds. A phase-field order parameter captures material degradation from ductile or brittle fracture, including evolving porosity from crazing. The theory is applied toward polymethyl methacrylate (PMMA) under intense dynamic loading. The high-pressure equilibrium response, with shear strength and temperature over known ranges, is well represented along the principal Hugoniot to pressures far exceeding shock decomposition. Predicted release wave velocities agree with experiment. A semi-analytical solution for steady waves describes the relatively lower-pressure viscoelastic setting, providing insight into relaxation times. One-dimensional calculations assess suitability of the model for representing spall fracture strengths seen in experiments over a range of initial temperatures and loading rates.

💡 Research Summary

This paper presents a comprehensive, geometrically nonlinear continuum‑mechanics framework capable of describing the full spectrum of dynamic phenomena observed in amorphous thermoplastic polymers, with a particular focus on polymethyl methacrylate (PMMA). The authors identify a gap in existing models: while many capture individual mechanisms such as viscoelasticity, plasticity, or fracture, none simultaneously incorporate thermo‑elastic response, rate‑dependent viscoelasticity, finite‑strain viscoplasticity, ductile and brittle fracture, glass‑transition effects, and shock‑induced chemical decomposition.

The theory is built on a thermodynamically consistent internal‑state‑variable (ISV) formulation. The deformation gradient is split into a total thermo‑elastic part (which includes both thermal expansion and viscoelastic deformation) and a residual plastic part that accounts for irreversible chain motions, free‑volume changes, and dilatational crazing. Unlike earlier multiplicative decompositions that treat instantaneous elasticity, viscoelasticity, and plasticity as separate factors, this approach embeds viscoelasticity within the thermo‑elastic component, simplifying the kinematics while preserving full rate‑form coupling through the velocity gradient.

Thermo‑elasticity is described using a logarithmic strain measure and a free‑energy function that depends on temperature, pressure, and the ISVs, yielding an equation of state valid up to >100 GPa. Viscoelasticity is modeled with tensor‑valued internal variables following the Holzapfel‑Simo framework, allowing nonlinear stress relaxation and capturing the strong pressure‑ and temperature‑dependence of bulk and shear moduli observed in PMMA.

Viscoplasticity employs both scalar (isotropic hardening/softening) and tensorial (plastic stretch) internal variables, enabling anisotropic evolution of yield surfaces and rate‑sensitive flow. The plastic flow rule incorporates temperature, pressure, and strain‑rate effects, and it is coupled to two scalar phase‑field order parameters that govern transitions across the glass‑transition temperature and the shock‑decomposition threshold. These order parameters can evolve smoothly or abruptly, depending on the chosen kinetic laws, thereby reproducing both gradual softening near Tg and the sudden volume collapse associated with shock‑induced chemical breakdown.

Fracture is treated with a regularized phase‑field model that distinguishes ductile shear yielding from brittle crazing. A scalar damage variable controls the degradation of the elastic stiffness, while an additional porosity variable accounts for the volumetric expansion accompanying craze formation. The fracture energy is linked to the evolving porosity, allowing the model to capture the large inelastic volume changes typical of crazing‑driven failure.

The complete constitutive set also includes a melting transition (via a glass‑transition order parameter) and a shock‑decomposition transition, each with its own kinetic law. Heat conduction and viscous dissipation are added through Fourier’s law and a Newtonian bulk viscosity term, ensuring realistic temperature evolution during high‑rate loading.

The framework is calibrated and validated against extensive experimental data for PMMA. High‑pressure Hugoniot calculations reproduce measured pressure–particle‑velocity relationships up to >120 GPa, including the 3.4 % volume collapse observed near 20–30 GPa, which the model attributes to the shock‑decomposition order parameter. Predicted release‑wave velocities match experimental values, confirming the accuracy of the thermo‑elastic EOS and the viscoelastic relaxation description.

A semi‑analytical solution for steady, low‑pressure shock waves provides insight into relaxation times and the rate dependence of the viscoelastic moduli. One‑dimensional spall simulations, performed over a range of initial temperatures and loading rates, capture the experimentally observed increase of spall strength with loading rate and its sharp decline across the glass‑transition temperature. The phase‑field damage variable successfully reproduces the transition from ductile shear‑band spall to brittle craze‑driven spall.

In conclusion, the authors deliver a unified, thermodynamically consistent model that simultaneously accounts for thermo‑elasticity, viscoelasticity, viscoplasticity, ductile and brittle fracture, glass‑transition, and shock‑induced decomposition. The model demonstrates excellent agreement with PMMA data across a vast range of pressures (0–120 GPa), strain rates (10⁻³–10⁶ s⁻¹), and temperatures (below and above Tg). Limitations include the extensive parameter identification required and the need for further multiscale validation. Future work is suggested on automated parameter calibration, extension to other amorphous polymers, and application to multilayered protective systems.